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Properties

Label 2-1350-5.2-c2-0-7
Degree $2$
Conductor $1350$
Sign $-0.608 - 0.793i$
Analytic cond. $36.7848$
Root an. cond. $6.06505$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + 2i·4-s + (5.44 + 5.44i)7-s + (2 − 2i)8-s − 3·11-s + (−2.32 + 2.32i)13-s − 10.8i·14-s − 4·16-s + (−4.34 − 4.34i)17-s + 22.6i·19-s + (3 + 3i)22-s + (−17.0 + 17.0i)23-s + 4.65·26-s + (−10.8 + 10.8i)28-s − 9.30i·29-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + 0.5i·4-s + (0.778 + 0.778i)7-s + (0.250 − 0.250i)8-s − 0.272·11-s + (−0.178 + 0.178i)13-s − 0.778i·14-s − 0.250·16-s + (−0.255 − 0.255i)17-s + 1.19i·19-s + (0.136 + 0.136i)22-s + (−0.740 + 0.740i)23-s + 0.178·26-s + (−0.389 + 0.389i)28-s − 0.320i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.608 - 0.793i$
Analytic conductor: \(36.7848\)
Root analytic conductor: \(6.06505\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1),\ -0.608 - 0.793i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6348707321\)
\(L(\frac12)\) \(\approx\) \(0.6348707321\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-5.44 - 5.44i)T + 49iT^{2} \)
11 \( 1 + 3T + 121T^{2} \)
13 \( 1 + (2.32 - 2.32i)T - 169iT^{2} \)
17 \( 1 + (4.34 + 4.34i)T + 289iT^{2} \)
19 \( 1 - 22.6iT - 361T^{2} \)
23 \( 1 + (17.0 - 17.0i)T - 529iT^{2} \)
29 \( 1 + 9.30iT - 841T^{2} \)
31 \( 1 - 8.04T + 961T^{2} \)
37 \( 1 + (4.77 + 4.77i)T + 1.36e3iT^{2} \)
41 \( 1 - 52.0T + 1.68e3T^{2} \)
43 \( 1 + (45.7 - 45.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (51.0 + 51.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (26.6 - 26.6i)T - 2.80e3iT^{2} \)
59 \( 1 + 68.3iT - 3.48e3T^{2} \)
61 \( 1 - 61.6T + 3.72e3T^{2} \)
67 \( 1 + (87.7 + 87.7i)T + 4.48e3iT^{2} \)
71 \( 1 + 88.4T + 5.04e3T^{2} \)
73 \( 1 + (62.4 - 62.4i)T - 5.32e3iT^{2} \)
79 \( 1 - 78.8iT - 6.24e3T^{2} \)
83 \( 1 + (-18.6 + 18.6i)T - 6.88e3iT^{2} \)
89 \( 1 + 50.5iT - 7.92e3T^{2} \)
97 \( 1 + (46.0 + 46.0i)T + 9.40e3iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.749576606811578539452870075640, −8.921864055383665522183466513823, −8.113440332404260116162053363012, −7.65012645639995648398508016250, −6.41434999632264782329819406405, −5.49446016635323147898372835644, −4.57918490596121375719445643866, −3.48416911977597981515360989811, −2.32909958729042507874298490569, −1.50383179527476599512305968263, 0.21325082772583066551736121508, 1.47135235014055696421039470546, 2.74341890662993674102046281576, 4.23901322192465169780087229569, 4.87188397908826669470577385503, 5.92073088845652032769297673348, 6.86975324770195891646149407158, 7.53086394458448780695454280576, 8.281823463739631023650117390472, 8.962738927084434906646419475215

Graph of the $Z$-function along the critical line